On metric tensor

In this thesis, Einstein's field equation is examined. All of the terms on the left side of this equation relate to the geometry of space, and all of the terms on the right side are physically related terms. These mathematical and physical terms in the equation are explained simply because it is primarily intended to understand this equation. In the following sections, important subjects of Riemannian geometry, Riemann Manifold and Pseudo-Riemann Manifold, Parallel transport, Connection and Covariant derivative, Curvature and Torsion, Levi-Civita Connections are mentioned. In addition, these subjects formed Einstein's field equation. In fact, the aim of this thesis is to explain at least what Einstein's field equations are mathematically in recent years when important discoveries about black holes have been made.